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History-Preserving Bisimilarity for Higher-Dimensional
Automata via Open Maps
Uli Fahrenberg, Axel Legay
To cite this version:
Uli Fahrenberg, Axel Legay. History-Preserving Bisimilarity for Higher-Dimensional Automata via
Open Maps. LICS 2013 - Twenty-Eighth Annual ACM/IEEE Symposium on Logic in Computer
Science, Jun 2013, New Orleans, United States. �hal-01087933�
History-Preserving Bisimilarity for
Higher-Dimensional Automata via Open Maps
Extended Abstract
Uli Fahrenberg
Axel Legay
Irisa / INRIA Rennes, France
One of the popular notions of equivalence for non-interleaving concurrent systems is history-preserving bisimilarity
(hp-bisimilarity). Higher-dimensional automata (HDA) [6], [7] is a non-interleaving formalism for reasoning about behavior of concurrent systems, which provides a generalization (up to hp-bisimilarity) to “the main models of concurrency proposed in the literature” [8].
Using open maps [4], we can show that hp-bisimilarity for HDA has a characterization directly in terms of (higher-dimensional) transitions of the HDA, rather than in terms of runs as e.g. for Petri nets. Our results imply decidability of hp-bisimilarity for finite HDA. They also put hp-bisimilarity firmly into the open-maps framework of [4] and tighten the connections between bisimilarity and weak topological
fibrations[1], [5].
A full version of this report is available as [3].
A precubical set is a graded set X = {Xn}n∈◆ together
with mappings δν
k : Xn → Xn−1, k = 1, . . . , n, ν = 0, 1,
satisfying the precubical identity δν kδ µ ℓ = δ µ ℓ−1δkν for k < ℓ. The mappings δν
k are called face maps, and elements of Xn
are called n-cubes. Faces δ0
kx of an element x ∈ X are to
be thought of as lower faces, δ1
kx as upper faces. Morphisms
f : X → Y of precubical sets are graded mappings f = {fn : Xn → Yn}n∈◆ which commute with the face maps:
δν
k◦fn= fn−1◦δνk. This defines a category pCub of precubical
sets and morphisms.
The category of higher-dimensional automata is the comma category HDA= ∗ ↓ pCub of pointed precubical sets and with morphisms which respect the point.
We say that a precubical set X is a path object if there is a (necessarily unique) sequence(x1, . . . , xm) of elements in X
such that xi6= xj for i6= j,
• for each x ∈ X there is j ∈ {1, . . . , m} for which
x= δν1
k1· · · δ
νp
kpxj for some indices ν1, . . . , νp and a
uniquesequence k1<· · · < kp, and
• for each j = 1, . . . , m − 1, there is k ∈ ◆ for which
xj = δk0xj+1 or xj+1= δk1xj.
If X and Y are path objects with representations(x1, . . . , xm),
(y1, . . . , yp), then a morphism f : X → Y is called a path extension if xj = yj for all j = 1, . . . , m (hence m ≤ p).
The category HDP of higher-dimensional paths (HDP) is the subcategory of HDA which as objects has pointed path objects, and whose morphisms are generated by isomorphisms and
pointed path extensions.
Following [2], we say that a morphism in HDA is open if it has the right lifting property with respect to HDP, and that HDA X, Y are bisimilar if there is Z ∈ HDA and a span of open maps X ← Z → Y in HDA. It can be shown [2] that X and Y are bisimilar iff n-cubes with matching lower faces can be matched; this is a straight-forward generalization of ordinary bisimulation for transition systems and appears hence to be rather badly suited for concurrent systems. We can, however, show that this bisimilarity is precisely hp-bisimilarity.
A cube path in a precubical set X is a morphism P → X from a path object P . Using the notion of adjacency from [7], [8], we can define what it means for two cube paths to be homotopic, i.e. to represent the same execution up to concurrency. The unfolding ˜X of a HDA X is then defined to be the set of homotopy classes of pointed cube paths in X. With suitable structure maps, this becomes a precubical set, indeed, a higher-dimensional tree.
The category of HDA up to homotopy HDAhhas as objects
HDA and as morphisms pointed precubical morphisms f : ˜
X → ˜Y of unfoldings. Noting that any HDP is isomorphic to its own unfolding, we have an embedding HDP ֒→ HDAh.
We can then say that a morphism in HDAhis homotopy open
if it has the right lifting property with respect to HDP and define homotopy bisimilarity accordingly.
Theorem: Two HDA are homotopy bisimilar iff they are hp-bisimilar [8], iff they are bisimilar.
Using an arrow category, we can easily extend the above considerations to the (more interesting) case of labeled HDA.
[1] Jiˇr´ı Ad´amek, Horst Herrlich, Jiˇr´ı Rosick´y, and Walter Tholen. Weak factorization systems and topological functors. Appl. Categ. Struct., 10(3):237–249, 2002.
[2] Uli Fahrenberg. A category of higher-dimensional automata. In
FOS-SACS, volume 3441 of LNCS, pages 187–201. Springer, 2005. [3] Uli Fahrenberg and Axel Legay. History-preserving bisimilarity for
higher-dimensional automata via open maps. CoRR, abs/1209.4927, 2012. http://arxiv.org/abs/1209.4927.
[4] Andr´e Joyal, Mogens Nielsen, and Glynn Winskel. Bisimulation from open maps. Inf. Comp., 127(2):164–185, 1996.
[5] Alexander Kurz and Jiˇr´ı Rosick´y. Weak factorizations, fractions and homotopies. Appl. Categ. Struct., 13(2):141–160, 2005.
[6] Vaughan Pratt. Modeling concurrency with geometry. In POPL, pages 311–322. ACM Press, 1991.
[7] Rob J. van Glabbeek. Bisimulations for higher dimensional automata. Email message, June 1991. http://theory.stanford.edu/∼rvg/hda. [8] Rob J. van Glabbeek. On the expressiveness of higher dimensional
